Path Cover Problems with Length Cost

For a graph G = ( V , E ) , a collection P of vertex-disjoint (simple) paths is called a path cover of G if every vertex v ∈ V is contained in exactly one path of P . The Path Cover problem (PC for short) is to find a minimum cardinality path cover of G . In this paper, we introduce generalizations of PC, where each path is associated with a weight (cost or profit). Our problem, Minimum (Maximum) Weighted Path Cover [MinPC (MaxPC)], is defined as follows: Let U = { 0 , 1 , ⋯ , n - 1 } . Given a graph G = ( V , E ) and a weight function f : U → R ∪ { + ∞ , - ∞ } that defines a weight for each path based on its length, the objective of MinPC (MaxPC) is to find a path cover P of G such that the total weight of the paths in P is minimized (maximized). Let L be a subset of U , and P L be the set of paths such that each path is of length ℓ ∈ L . We consider Min P L PC with binary cost, i.e., the cost function is f ( ℓ ) = 1 if ℓ ∈ L ; otherwise, f ( ℓ ) = 0 . We also consider Max P L PC with f ( ℓ ) = ℓ + 1 , if ℓ ∈ L ; otherwise, f ( ℓ ) = 0 . Many well-known graph theoretic problems such as the Hamiltonian Path and the Maximum Matching problems can be modeled using Min P L PC and Max P L PC. In this paper, we first show that deciding whether Min P { 0 , 1 , 2 } PC has a 0-weight solution is NP-complete for planar bipartite graphs of maximum degree three, and consequently, (i) for any constant σ ≥ 1 , there is no polynomial-time approximation algorithm with approximation ratio σ for Min P { 0 , 1 , 2 } PC unless P = NP, and (ii) Max P { 3 , ⋯ , n - 1 } PC is NP-hard for the same graph class. Next, we present a polynomial-time algorithm for Min P { 0 , 1 , ⋯ , k } PC on graphs with bounded treewidth for a fixed k . Lastly, we present a 4-approximation algorithm for Max P { 3 , ⋯ , n - 1 } PC, which becomes a 2.5-approximation algorithm for subcubic graphs.